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NAME
DATE
4-3
PERIOD
Study Guide and Intervention
Solving Quadratic Equations by Factoring
Factored Form
To write a quadratic equation with roots p and q, let (x - p)(x - q) = 0.
Then multiply using FOIL.
Example
Write a quadratic equation in standard form with the given roots.
7 1
,−
b. - −
a. 3, -5
8 3
(x - p)(x - q) = 0
(x - 3)[x - (-5)] = 0
(x - 3)(x + 5) = 0
x2 + 2x - 15 = 0
(x - p)(x - q) = 0
Write the pattern.
Replace p with 3, q with -5.
Simplify.
x - (-−78 ) (x - −13 ) = 0
(x + −78 )(x - −13 )
Use FOIL.
The equation x2 + 2x - 15 = 0 has roots
3 and -5.
(8x + 7)
8
=0
(3x - 1)
3
−− =0
24 (8x + 7)(3x - 1)
−−
24
= 24 0
24x2 + 13x - 7 = 0
The equation 24x2 + 13x - 7 = 0 has
7
1
roots - −
and −
.
8
3
Write a quadratic equation in standard form with the given root(s).
1. 3, -4
2
x + x - 12 = 0
3. 1, 9
2. -8, -2
2
x + 10x + 16 = 0
5. 10, 7
4. -5
2
x + 10x + 25 = 0
1
7. - −
,5
3
2
3x - 14x - 5 = 0
2
10. 3, −
5x - 17x + 6 = 0
2
2
, -−
13. −
3
2
9x - 4 = 0
7 7
,−
16. - −
8 2
2
16x - 42x - 49 = 0
Chapter 4
x - 17x + 70 = 0
3
4
2
3x - 8x + 4 = 0
2
4x + 25x - 21 = 0
1
12. 9, −
6
2
9x + 13x + 4 = 0
5
1
14. −
, -−
4
x 2 - 13x - 30 = 0
3
9. -7, −
2
8. 2, −
9
2
3
6. -2, 15
2
4
11. - −
, -1
5
x 2 - 10x + 9 = 0
6x 2 - 55x + 9 = 0
3 1
15. −
,−
7 5
2
2
8x - 6x - 5 = 0
1 3
17. −
,−
35x 2 - 22x + 3 = 0
1 1
18. −
,−
2 4
8 6
2
8x - 10x + 3 = 0
17
48x 2 - 14x + 1 = 0
Glencoe Algebra 2
Lesson 4-3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
NAME
4-3
DATE
PERIOD
Study Guide and Intervention
(continued)
Solving Quadratic Equations by Factoring
Solve Equations by Factoring
When you use factoring to solve a quadratic equation,
you use the following property.
Zero Product Property
Example
For any real numbers a and b, if ab = 0, then either a = 0 or b =0, or both a and b = 0.
Solve each equation by factoring.
a. 3x2 = 15x
b. 4x2 - 5x = 21
4x2 - 5x = 21 Original equation
4x2 - 5x - 21 = 0 Subtract 21 from both sides.
(4x + 7)(x - 3) = 0 Factor the trinomial.
4x + 7 = 0 or x - 3 = 0 Zero Product Property
3x2 = 15x Original equation
3x2 - 15x = 0
Subtract 15x from both sides.
3x(x - 5) = 0
Factor the binomial.
3x = 0 or x - 5 = 0
Zero Product Property
x = 0 or
x=5
Solve each equation.
7
x = -−
or
x=3
4
The solution set is {0, 5}.
{
Solve each equation.
}
7
The solution set is - −
,3 .
4
Exercises
Solve each equation by factoring.
1. 6x2 - 2x = 0
0, −13 0, −76 7. x2 + x - 30 = 0
{5, -6}
10. 4x2 + 27x - 7 = 0
−14 , -7
13. 12x2 - 8x + 1 = 0
−16 , −12 16. 2x2 - 11x - 40 = 0
8, - −52 19. 8x2 - 14x + 3 = 0
−32 , −14 22. 12x2 + 25x + 12 = 0
- −43 , - −34 Chapter 4
3. 20x2 = -25x
0, - −54 {0, 7}
5. 6x2 - 27x = 0
0, −92 6. 12x2 - 8x = 0
0, −23 8. 2x2 - x - 3 = 0
−32 , -1
11. 3x2 + 29x - 10 = 0
-10, −13 14. 5x2 + 28x - 12 = 0
−25 , -6
17. 2x2 + 21x - 11 = 0
-11, −12 20. 6x2 + 11x - 2 = 0
-2, −16 23. 12x2 + 18x + 6 = 0
-−12 , -1
18
9. x2 + 14x + 33 = 0
{-11, -3}
12. 6x2 - 5x - 4 = 0
- −12 , −43 15. 2x2 - 250x + 5000 = 0
{100, 25}
18. 3x2 + 2x - 21 = 0
−73 , -3
21. 5x2 + 17x - 12 = 0
−35 , -4
24. 7x2 - 36x + 5 = 0
−17 , 5
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4. 6x2 = 7x
2. x2 = 7x